3.218 \(\int \frac{x}{\sqrt{-4-12 x-9 x^2}} \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{9} \sqrt{-9 x^2-12 x-4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{-9 x^2-12 x-4}} \]

[Out]

-Sqrt[-4 - 12*x - 9*x^2]/9 - (2*(2 + 3*x)*Log[2 + 3*x])/(9*Sqrt[-4 - 12*x - 9*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0119671, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 608, 31} \[ -\frac{1}{9} \sqrt{-9 x^2-12 x-4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{-9 x^2-12 x-4}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

-Sqrt[-4 - 12*x - 9*x^2]/9 - (2*(2 + 3*x)*Log[2 + 3*x])/(9*Sqrt[-4 - 12*x - 9*x^2])

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{-4-12 x-9 x^2}} \, dx &=-\frac{1}{9} \sqrt{-4-12 x-9 x^2}-\frac{2}{3} \int \frac{1}{\sqrt{-4-12 x-9 x^2}} \, dx\\ &=-\frac{1}{9} \sqrt{-4-12 x-9 x^2}+-\frac{(2 (-6-9 x)) \int \frac{1}{-6-9 x} \, dx}{3 \sqrt{-4-12 x-9 x^2}}\\ &=-\frac{1}{9} \sqrt{-4-12 x-9 x^2}-\frac{2 (2+3 x) \log (2+3 x)}{9 \sqrt{-4-12 x-9 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0075156, size = 35, normalized size = 0.73 \[ \frac{(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

((2 + 3*x)*(2 + 3*x - 2*Log[2 + 3*x]))/(9*Sqrt[-(2 + 3*x)^2])

________________________________________________________________________________________

Maple [A]  time = 0.116, size = 31, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2+3\,x \right ) \left ( -3\,x+2\,\ln \left ( 2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-(2+3*x)^2)^(1/2),x)

[Out]

-1/9*(2+3*x)*(-3*x+2*ln(2+3*x))/(-(2+3*x)^2)^(1/2)

________________________________________________________________________________________

Maxima [C]  time = 1.71448, size = 28, normalized size = 0.58 \begin{align*} -\frac{1}{9} \, \sqrt{-9 \, x^{2} - 12 \, x - 4} - \frac{2}{9} i \, \log \left (x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-(2+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-9*x^2 - 12*x - 4) - 2/9*I*log(x + 2/3)

________________________________________________________________________________________

Fricas [C]  time = 1.61575, size = 42, normalized size = 0.88 \begin{align*} -\frac{1}{3} i \, x + \frac{2}{9} i \, \log \left (x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-(2+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*I*x + 2/9*I*log(x + 2/3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (3 x + 2\right )^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-(2+3*x)**2)**(1/2),x)

[Out]

Integral(x/sqrt(-(3*x + 2)**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-(2+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

undef